一个耦合非光滑动力系统：全局适定性、稳定性和灵敏度分析
A Coupled Nonsmooth Dynamical System: Global Well-Posedness, Stability and Sensitivity Analysis
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中文总结 AI 辅助
研究耦合非光滑动力系统，通过变换变量将代数律转化为变分不等式，在特定假设下分析其性质，结合多种方法证明解的全局存在唯一性等，推导稳定性准则并通过实例说明相关问题。
中文摘要 AI 辅助
本文研究了一个耦合非光滑动力系统，其中一个半线性发展方程与一个由依赖于时间的凸集的法锥控制的隐式代数律相耦合。主要困难在于代数变量不是由显式反馈给出，而是必须从依赖于状态的法锥关系中恢复。通过使用一个变换变量，我们将每个冻结的代数律重铸为凸集上的变分不等式。在强伪单调、Lipschitz连续假设下，这个冻结问题有唯一的代数响应，并且在没有投影收缩论证的情况下允许对状态到控制映射进行灵敏度估计。我们还给出了该假设的可验证充分条件，包括加权强单调构造和标准Lipschitz小条件，并指出该框架允许强伪单调非单调冻结算子。然后，耦合系统被简化为一个具有单个状态变量的半线性发展方程。通过将\(C_0\)半群估计与Bielecki不动点论证相结合，我们证明了有限时间区间上温和解对的全局存在性和唯一性，并通过显式连续依赖性估计建立了Hadamard适定性。我们进一步推导了耗散半群状态下的增量指数稳定性准则，并证明了参数到解映射关于初始数据和外部参数的连续性。一个简化的接触力学例子说明了变分不等式的表述和一个显式投影残差，该残差可作为离散化后投影和牛顿型内部求解器的起点。
英文摘要
This paper studies a coupled nonsmooth dynamical system in which a semilinear evolution equation is coupled with an implicit algebraic law governed by the normal cone to a time-dependent convex set. The main difficulty is that the algebraic variable is not given by an explicit feedback, but must be recovered from a state-dependent normal cone relation. Using a transformed variable, we recast each frozen algebraic law as a variational inequality over a convex set. Under a strongly pseudomonotone, Lipschitz continuous hypothesis, this frozen problem has a unique algebraic response and admits a sensitivity estimate for the state-to-control map without a projection-contraction argument. We also give verifiable sufficient conditions for this hypothesis, including a weighted strongly monotone construction and a standard Lipschitz-smallness condition, and point out that the framework allows strongly pseudomonotone nonmonotone frozen operators. The coupled system is then reduced to a semilinear evolution equation with a single state variable. By combining $C_0$-semigroup estimates with a Bielecki fixed-point argument, we prove global existence and uniqueness of mild solution pairs on finite time intervals and establish Hadamard well-posedness through explicit continuous-dependence estimates. We further derive an incremental exponential stability criterion in the dissipative semigroup regime and prove continuity of the parameter-to-solution map with respect to the initial datum and an external parameter. A reduced contact-mechanics example illustrates the variational inequality formulation and an explicit projection residual that can be used as a starting point for projection- and Newton-type inner solvers after discretization.
